Unique Continuation for Stochastic Hyperbolic Equations Qi Lu¨∗ and Zhongqi Yin† 7 1 0 2 n Abstract a J In this paper, we derive a local unique continuation property for stochastic hyperbolic 3 equations without boundary conditions. This result is proved by a global Carleman 1 estimate. ] P A 2000 Mathematics Subject Classification. Primary 60H15, 93B07. . h t a Key Words. Stochastic hyperbolic equation, Carleman estimate, unique continuation. m [ 1 1 Introduction v 9 9 Let T > 0, G ∈ Rn (n ∈ N) be a given bounded domain. Throughout this paper, we will 5 use C to denote a generic positive constant depending only on T, G, which may change from 3 0 line to line. Denote Q = G×(0,T). 1. Let (Ω,F,F,P) with F =△ {F } be a complete filtered probability space on which t t≥0 0 a one dimensional standard Brownian motion {W(t)} is defined. Assume that H is a 7 t≥0 1 Fréchet space. Let L2(0,T;H) be the Banach space consisting all H-valued F-adapted F v: processX(·)suchthatE|X(·)|2 < +∞,onwhichthecanonicalquasi-normisendowed. i L2F(0,T;H) X By L∞(0,T;H) we denote the Fréchet space of all H-valued F-adapted bounded processes F r equipped with the canonical quasi-norm and by L2(Ω;C([0,T];H)) the Fréchet space of all a F H-valued F-adapted continuous precesses X(·) with E|X(·)|2 < +∞ and equipped C([0,T];H) with the canonical quasi-norm. Thispaperisdevotedtothestudyofalocaluniquecontinuationpropertyforthefollowing stochastic hyperbolic equation: σdz −∆zdt = b z +b ·∇z +b z dt+b zdW(t) in Q, (1.1) t 1 t 2 3 4 ∗School of Mathematics, Sichuan(cid:0)University, Chengdu 6(cid:1)10064, Sichuan Province, China. The author is partiallysupportedbyNSFofChinaundergrants11471231,theFundamentalResearchFundsfortheCentral UniversitiesinChinaundergrant2015SCU04A02andGrantMTM2011-29306-C02-00oftheMICINN,Spain. E-mail: lu@scu.edu.cn. †Department of Mathematics, Sichuan Normal University, Chengdu 610068, P. R. China. E-mail: zhongqiyin@sicnu.edu.cn. 1 where, σ ∈ C1(Q) is positive, b ∈ L∞(0,T;L∞(G)), b ∈ L∞(0,T;L∞(G;Rn)), 1 F loc 2 F loc b ∈ L∞(0,T;Ln (G)), b ∈ L∞(0,T;L∞(G)). 3 F loc 4 F loc Put H =△ L2(Ω;C([0,T];H1 (G)))∩L2(Ω;C1([0,T];L2 (G))). (1.2) T F 0,loc F loc The definition of solutions to equation (1.1) is given in the following sense. Definition 1.1 We call z ∈ H a solution of equation (1.1) if for each t ∈ [0,T], G′ ⊂⊂ G T and η ∈ H1(G′), it holds that 0 z (t,x)η(x)dx− z (0,x)η(x)dx t t ZG′ ZG′ t 1 = −∇z(s,x)·∇η(x)+ b z +b ·∇z +b z η(x) dxds (1.3) 1 t 2 3 σ Z0 ZG′ t h1 (cid:0) (cid:1) i + b zη(x)dxdW(s), P-a.s. 4 σ Z0 ZG′ Let S ⊂⊂ G be a C2-hypersurface. Let x ∈ S \∂G and suppose as well that S divides 0 the ball B (x ) ⊂ G, centered at x and with radius ρ, into two parts D+ and D−. Denote ρ 0 0 ρ ρ as usual by ν(x) the unit normal vector to S at x inward to D+. ρ Let y be a solution to equation (1.1). Let ε > 0. This paper is devoted to the following local unique continuation problem: (Pu) Can y in D+×(ε,T −ε) be uniquely determined by the values of y in D−×(0,T)? ρ ρ In other words, Problem (Pu) concerns that whether the values of the solution in one side of S uniquely determines its values in the other side. Clearly, it is equivalent to the following problem: (Pu1) Can we conclude that y = 0 in D+×(ε,T−ε), provided that y = 0 in D−×(0,T)? ρ ρ Unique continuation problems for deterministic PDEs are studied extensively in the literature. Generally speaking, a unique continuation result is a statement in the following sense: Let u be a solution of a PDE and two regions M ⊂ M . Then u is determined uniquely 1 2 in M by its values in M . 2 1 Problem (Pu) is a natural generalization of the unique continuation problems under the stochastic setting, i.e., M = D− and M = D. 1 2 There is a long history for the study of unique continuation property (UCP for short) for deterministic PDEs. Classical results dates back to Cauchy-Kovalevskaya theorem and Holmgren’s uniqueness theorem. These two results need the coefficients of the PDE to be analytic to get the UCP. In 1939, T. Carleman introduced in [5] a new method, which was based on weighted estimates, to prove UCP for two dimensional elliptic equations with L∞ coefficients. This method, which is called “Carleman estimates method” nowadays, turned out to be a quite powerful tools and has been developed extensively in the literature and 2 become the most useful tool to obtainUCP forPDEs (e.g. [6, 8, 9, 23, 24, 28]). In particular, unique continuation results for solutions of hyperbolic equations across hypersurfaces were studied by many authors (e.g. [7, 10, 19, 20, 22]). Compared with the deterministic PDEs, there are very few results concerning UCP for stochastic PDEs. To our best knowledge, [25] is the first result for UCP of stochastic PDEs, in which the author shows that a solution to a stochastic parabolic equation vanishes, provided that it vanishes in any subdomain and this result was improved in [11, 16] where less geometric condition is assumed to the set where the solution vanishes. The first result of UCP for stochastic hyperbolic equations was obtain in [26]. Some improvement was done in [15, 17]. The results in [26] and [15] are concerning the global UCP for stochastic hyperbolic equations with a homogeneous Dirichlet boundary condition, i.e., they conclude that the solution to a stochastic hyperbolic equation vanishes, provided that it equals zero in a large enough subdomain. In this paper, we focus on the local UCP for stochastic hyperbolic equations without boundary condition, that is, can a solution be determined locally? To present the main result of this paper, let us first introduce the following notion. Definition 1.2 Let x ∈ S and K > 0. S is said to satisfy the outer paraboloid condition 0 with K at x if there exists a neighborhood V of x and a paraboloid P tangential to S at x 0 0 0 and P ∩V ⊂ D− with P congruent to x = K n x2. ρ 1 j=2 j P The main result in this paper is the following one. Theorem 1.1 Let x ∈ S \∂S such that ∂σ(x0,0) < 0, and let S satisfy the outer paraboloid 0 ∂ν condition with −∂σ(x ,0) K < ∂ν 0 . (1.4) 4(|σ|L∞(Bρ(x0,0)) +1) Let z ∈ H be a solution of the equation (1.1) satisfying that T ∂z z = = 0 on (0,T)×S, P-a.s. (1.5) ∂ν Then, there is a neighborhood V of x and ε ∈ (0,T/2) such that 0 z = 0 in (V ∩D+)×(ε,T −ε), P-a.s. (1.6) ρ Remark 1.1 In Theorem 1.1, we assume that ∂σ(x0,0) < 0. This is a reasonable assumption ∂ν since the UCP may not hold if it is not fulfilled (e.g. [1]). It can be regarded as a kind of pseudoconvex condition (e.g. [8, Chapter XXVII]). Remark 1.2 If S is a hyperplane, then Condition 1.7 always satisfies since we can take K = 0. Remark 1.3 From Theorem 1.1, one can get many classical UCP results for deterministic hyperbolic equations (e.g. [7, 18, 22]). As an immediate corollary of Theorem 1.1, we have the following UCP. 3 Corollary 1.1 Let x ∈ S \∂S such that ∂σ(x0,0) < 0, and let S satisfy the outer paraboloid 0 ∂ν condition with −∂σ(x ,0) K < ∂ν 0 . (1.7) 4(|σ|L∞(Bρ(x0,0)) +1) Then for any z ∈ H solve equation (1.1) satisfying that T z = 0 on D− ×(0,T), P-a.s. , (1.8) ρ there is an ε ∈ (0,T/2) such that z = 0 in D+ ×(ε,T −ε), P-a.s. (1.9) ρ Similar to the deterministic settings, we shall use the stochastic versions of Carleman estimate for stochastic hyperbolic equations to establish our estimate. Carleman estimates for stochastic PDEs are studied extensively in recent years (see [4, 12, 13, 14, 15, 21, 27] and the reference therein). Carleman estimate for stochastic hyperbolic equations was first obtained in [26]. Compared with the result in [26], we need to handle a more complex case (see Section 2 for more details). The rest of this paper is organized as follows. In Section 2, we derive a point-wise estimate for stochastic hyperbolic operator, which is the key tool to establish the desired Carleman estimate in this paper. In Section 3, we explain the choice of weight function in the Carleman estimate. In Section 4 is devoted to the proof of a Carleman estimate while Section 5 is addressed to the proof of the main result. 2 A point-wise estimate for stochastic hyperbolic operator Weintroducethefollowingpoint-wiseCarlemanestimateforstochastic hyperbolicoperators. This estimate has its own independent interest and will be of particular importance in the proof for the main result. Lemma 2.1 Let ℓ,Ψ ∈ C2((0,T)×Rn). Assume u is an H2 (Rn)-valued F-adapted process loc such that u is an L2(Rn)-valued semimartingale. Set θ = eℓ and v = θu. Then, for a.e. t x ∈ Rn and P-a.s. ω ∈ Ω, n n θ −2σℓ v +2 bijℓ v +Ψv σdu − (biju ) dt t t i j t i j (cid:16) iX,j=1 (cid:17)h iX,j=1 i n n + 2bijbi′j′ℓi′vivj′ −bijbi′j′ℓivi′vj′ −2bijℓtvivt +σbijℓivt2 iX,j=1hi′X,j′=1(cid:16) (cid:17) n 1 +Ψbijv v − Aℓ + Ψ bijv2 dt+d σ bijℓ v v i i i t i j 2 j (cid:16) (cid:17) i h iX,j=1 n 1 −2σ bijℓ v v +σ2ℓ v2 −σΨv v + σAℓ + (σΨ) v2 (2.1) i j t t t t t 2 t iX,j=1 (cid:16) (cid:17) i 4 n n = (σ2ℓ ) + (σbijℓ ) −σΨ v2 −2 [(σbijℓ ) +bij(σℓ ) ]v v t t i j t j t t j i t (cid:26)h iX,j=1 i iX,j=1 n n + (σbijℓt)t + 2bij′(bi′jℓi′)j′ −(bijbi′j′ℓi′)j′ +Ψbij vivj Xi,j=h i′X,j′=1(cid:16) (cid:17) i n 2 +Bv2 + −2σℓ v +2 bijℓ v +Ψv dt+σ2θ2ℓ (du )2, t t i j t t (cid:16) iX,j=1 (cid:17) (cid:27) where (du )2 denotes the quadratic variation process of u , and A and B are stated as follows: t t n A =△ σ(ℓ2 −ℓ )− (bijℓ ℓ −bijℓ −bijℓ )−Ψ, t tt i j j i ij  i,j=1 X (2.2)  B =△ AΨ+(σAℓ ) − (Abijℓ ) + 1 (σΨ) − n (bijΨ ) . t t i j tt i j 2  Xi,j h iX,j=1 i     Remark 2.1 When σ = 1, equality (2.1) had been established in [26]. The computation for the general σ is more complex. One needs to handle the terms concerning σ carefully. Proof of Lemma 2.1. By v(t,x) = θ(t,x)u(t,x), we have u = θ−1(v −ℓ v),u = θ−1(v −ℓ v) t t t j j j for j = 1,2,...,n. Then, for that θ is deterministic, we have σdu = σd[θ−1(v −ℓ v)] = σθ−1 dv −2ℓ v dt+(ℓ2 −ℓ )vdt . (2.3) t t t t t t t tt h i Moreover, we find that n n (biju ) = bijθ−1(v −ℓ v) i j i i j iX,j=1 iX,j=1(cid:16) (cid:17) (2.4) n = θ−1 (bijv ) −2bijℓ v +(bijℓ ℓ −bijℓ −bijℓ )v . i j i j i j j i ij iX,j=1h i As an immediate result of (2.3) and (2.4), we have that n σdu − (biju ) dt t i j i,j=1 X n n = θ−1 σdv − (bijv ) dt + −2σℓ v +2 bijℓ v dt (2.5) t i j t t i j h(cid:16) iX,j=1 (cid:17) (cid:16) iX,j=1 (cid:17) n + σ(ℓ2 −ℓ )− (bijℓ ℓ −bijℓ −bijℓ ) vdt . t tt i j j i ij (cid:16) iX,j=1 (cid:17) i 5 Therefore, by (2.5) and the definition of A in (2.2), we get n n θ −2σℓ v +2 bijℓ v +Ψv σdu − (biju ) dt t t i j t i j (cid:16) iX,j=1 (cid:17)(cid:16) iX,j=1 (cid:17) n = −2σ2ℓ v +2σ bijℓ v +σΨv dv t t i j t (cid:16) iX,j=1 (cid:17) (2.6) n n + −2σℓ v +2 bijℓ v +Ψv − (bijv ) +Av dt t t i j i j (cid:16) iX,j=1 (cid:17)(cid:16) iX,j=1 (cid:17) n 2 + −2σℓ v +2 bijℓ v +Ψv dt. t t i j (cid:16) iX,j=1 (cid:17) Let us continue to analyze the first two terms in the right-hand side of (2.6). For the first term in the right-hand side of (2.6), we find that −2σ2ℓ v dv = d(−σ2ℓ v2)+σ2ℓ (dv )2 +(σ2ℓ ) v2dt, t t t t t t t t t t n n n n 2σ bijℓ v dv = d 2σv bijℓ v −2 (σbijℓ ) v v dt−2σ bijℓ v v dt,  i j t t i j i t j t i jt t   iX,j=1 (cid:16) iX,j=1 (cid:17) iX,j=1 iX,j=1 σΨvdv = d(Ψσvv )−(σΨ) vv dt−σΨv2dt. t t t t t     Therefore, we get that n −2σℓ v +2 bijℓ v +Ψv σdv t t i j t (cid:16) iX,j=1 (cid:17) n 1 = d −σ2ℓ v2 +2σv bijℓ v +σΨvv − (σΨ) v2 t t t i j t 2 t (cid:16) iX,j=1 (cid:17) (2.7) n n − (σbijℓ v2) − (σℓ ) + (σbijℓ ) −σΨ v2 i t j t t i j t hiX,j=1 (cid:16) iX,j=1 (cid:17) n 1 + 2 (σbijℓ ) v v − (σΨ) v2 dt+σ2ℓ (dv )2. i t j t tt t t 2 iX,j=1 i In a similar manner, for the second term in the right-hand side of (2.6), we find that n −2σℓ v − (bijv ) +Av t t i j h iX,j=1 i n n n = 2 (σbijℓ v v ) − bij(σℓ ) v v + (σbijℓ ) v v (2.8) t i t j t j i t t t i j hiX,j=1 iX,j=1 i iX,j=1 n − σ bijℓ v v +σAℓ v2 +(σAℓ ) v2, t i j t t t t (cid:16) iX,j=1 (cid:17) 6 n n 2 bijℓ v − (bijv ) +Av i j i j iX,j=1 h iX,j=1 i n n = − 2bijbi′j′ℓi′vivj′ −bijbi′j′ℓivi′vj′ −Abijℓiv2 (2.9) j iX,j=1hi′X,j′=1(cid:16) (cid:17) i n n + 2bij′(bi′jℓi′)j′ −(bijbi′j′ℓi′)j′ vivj − (Abijℓi)jv2, i,j,Xi′,j′=1h i iX,j=1 and n n n 1 Ψv − (bijv ) +Av = − Ψbijvv − Ψ bijv2 +Ψ bijv v i j i i i j 2 j h iX,j=1 i iX,j=1(cid:16) (cid:17) iX,j=1 (2.10) n 1 + − (bijΨ ) +AΨ v2. i j 2 h iX,j=1 i Finally, from (2.6) to (2.10), we arrive at the desired equality (2.1). 3 Choice of the weight function In this section, we explain the choice of the weight function which will be used to establish our global Carleman estimate. Although such kind of functions are already used in [2]. We give full details for it for the sake of completion and the convenience of readers. The weight function is in the following form: n 1 1 T 2 1 ϕ(x,t) = hx + x2 + t− + τ, (3.1) 1 2 j 2 2 2 Xj=2 (cid:16) (cid:17) where h and τ are suitable parameters, whose precise meanings will be explained in the sequel. Without loss of generality, we assume that 0 = (0,··· ,0) ∈ S\∂S and ν(0) = (1,··· ,0). For some r > 0, for that S is C2, we can parameterize S in the neighborhood of the origin by x = γ(x ,··· ,x ), |x |2 +···+|x |2 < r. (3.2) 1 2 n 2 n For brevity of notations, denote ∂σ a(x,t) = . ∂ν Hereafter, we set T T 2 B 0, = (x,t) : (x,t) ∈ Rn+1, |x|2 + t− < r2 , r 2 2 (3.3)  (cid:26) (cid:27) (cid:16) (cid:17) (cid:16) (cid:17) Br(0) = {x : x ∈ Rn, |x| < r}.   7 By (1.7), we have that −α = a(0,0) < 0, 0  α 0 K < ,   4(|σ|L∞(Br(0,T/2)) +1) (3.4)   n n  −K x2 < γ(x ,··· ,x ), if x2 < r. j 2 n j   j=2 j=2  X X    Let  M1 = max |σ|C1(Br(0,0)),1 . (3.5) Denote (cid:8) (cid:9) D− = {x : x ∈ B (0), x < γ(x ,··· ,x )}, D+ = B (0)\D−. r r 1 2 n r r r For any α ∈ (0,α ), in accordance with the continuity of a(x,t) and the first inequality in 0 (3.4), it is clear that there exists a δ > 0 small enough such that 0 < δ < min{1,r2}, which 0 0 would be specified later, and 2 T a(x,t) < −α if |x|2 + t− ≤ δ . (3.6) 0 2 (cid:18) (cid:19) Letting M0 = |σ|L∞(Br(0,T/2)), by the second inequality in (3.4), we can always choose N > 0 so large that 1 α K < < . (3.7) 2h 4(M +1) 0 Following immediately from (3.7), we have that 1−2hK > 0, hα−2(M +1) > 0. (3.8) 0 For K and h such chosen, we will further take τ ∈ (0,1) so small that K 1 2 2τ max , τ2 + ≤ δ . (3.9) 0 1−2hK 2h 1−2hK (cid:12) n o(cid:12) (cid:12) (cid:12) For convenience of notations, by denoting by µ (τ) the term in the left hand side of (3.9) (cid:12) (cid:12) 0 and letting A = min{σ,1}, we further assume that 0 h2A > 2hM µ (τ)+2M µ (τ), 0 1 0 1 0 (3.10)  αh > 2(M2 +pM ) µ (τ)−(M2 +nM )−(n−1).  1 1 0 0 0 For any positive number µ with 2µp> τ, let N Q = (x,t) ∈ Rn+1 | x > γ(x ,x ,··· ,x ), x2 < δ ,ϕ(x,t) < µ . (3.11) µ 1 2 3 n j 0 n Xj=2 o The set Q defined in this style is not empty. It is only to prove that the defining conditon τ ϕ(x,t) < µ is compatible with the first defining condition, i.e., x > γ(x ,x ,··· ,x ). By 1 2 3 n 8 assumption, we know that γ(x ,x ,··· ,x ) > −K n x2, then together with the first 2 3 n j=2 j inequality in (3.8), we have that P n n 2 1 1 T 1 ϕ(x,t) ≥ −hK x2 + x2 + t− + τ j 2 j 2 2 2 j=2 j=2 (cid:18) (cid:19) X X n 1 1 T 2 1 = −Kh x2 + t− + τ 2 j 2 2 2 (cid:16) (cid:17)Xj=2 (cid:16) (cid:17) τ > . 2 Noting that (x,t) ∈ Q implies ϕ(x,t) < µ, together with 2µ > τ, we see by definition that µ Q 6= ∅ as desired. µ In what follows, we will show that how to determine the number δ appearing in (3.9). 0 Let (x,t) ∈ Q . From the definition of Q and noting that γ(x ,x ,··· ,x ) > −K n x2, τ τ 2 3 n j=2 j we find that n P −τ 1 T 2 τ τ x ≤ x2 − t− + ≤ . (3.12) 1 2h j 2h 2 2h 2h Xj=2 (cid:16) (cid:17) n On the other hand, by −K x2 ≤ x , we find that j 1 j=2 X n n 1 1 T 2 1 −Kh x2 + x2 + t− + τ ≤ τ. j 2 j 2 2 2 Xj=2 Xj=2 (cid:16) (cid:17) Thus n τ x2 < . j 1−2Kh j=2 X We then get that n Kτ −x ≤ K x2 < . (3.13) 1 j 1−2Kh j=2 X Combining (3.12) and (3.13), we arrive at K 1 |x | ≤ max , τ. (3.14) 1 1−2hK 2h (cid:26) (cid:27) Thus, by the restriction imposed on ϕ(x,t) in the definition of Q and (3.13), we find τ that n 1 1 T 2 τ τ > ϕ(x,t) = hx + x2 + t− + 1 2 j 2 2 2 Xj=2 (cid:16) (cid:17) (3.15) n Khτ 1 1 T 2 τ > − + x2 + t− + . 1−2Kh 2 j 2 2 2 Xj=2 (cid:16) (cid:17) 9 This gives that T 2 2Khτ τ t− < +τ = . (3.16) 2 1−2Kh 1−2Kh (cid:16) (cid:17) Correspondingly, we have that n T 2 T 2 K 1 2 2τ |x|2 + t− = x2 + x2 + t− ≤ max , τ2 + . 2 1 j 2 1−2Kh 2h 1−2Kh (cid:16) (cid:17) Xj=2 (cid:16) (cid:17) (cid:12) (cid:26) (cid:27)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) Returning back to (3.6), by (3.13), (3.14) and (3.16), we choose the δ in the following 0 style: K 1 2 2τ δ > µ (τ) = max , τ2 + . (3.17) 0 0 1−2Kh 2h 1−2Kh (cid:26) (cid:27) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 4 A global Carleman estimate This section is devoted to establishing a global Carleman estimate for the stochastic hyperbolic operator presented in Section 1, based on the point-wise Carleman estimate given in Section 2. It will be shown that it is the key to the proof of the main result. We have the following global Carleman estimate. Theorem 4.1 Let u be an H2 (Rn)-valued F-adapted process such that u is an L2(Rn)- loc t valued semimartingale. If u is supported in Q , then there exist a constant C depending on τ b ,i = 1,2,3 and a s > 0 depending on σ,τ such that for all s ≥ s it holds that i 0 0 E θ −2σℓ v +2∇ℓ·∇v σdu −∆udt dx t t t ZQτ (cid:0) (cid:1)(cid:0) (cid:1) ≥ CE sλ2ϕ−λ−2(|∇v|2+v2)+s3λ4ϕ−3λ−4v2 dxdt (4.1) t ZQτ h i +E −2σℓ v +2∇ℓ·∇v 2dxdt+CE σ2θ2ℓ (du )2dx. t t t t ZQτ ZQτ (cid:0) (cid:1) Proof. We apply the result of Lemma 2.1 to show our key Carleman estimate. Let (bij) = I , the unit matrix of nth order and ϕ = 0. Then, from (2.1), we find that 1≤i,j≤n n θ(−2σℓ v +2∇ℓ·∇v)[σdu −∆udt] t t t + ∇· 2(∇v·∇ℓ)∇v −|∇v|2∇ℓ−2ℓ v ∇v +σv2∇ℓ−A∇ℓv2 dt t t t h i + d σℓt|∇v|2 −2σ∇ℓ·∇vvt +σ2ℓtvt2 +σAℓtv2 (4.2) h i = (σ2ℓ ) +∇·(σ∇ℓ) v2−2 (σ∇ℓ) +∇(σℓ ) ·∇vv + (σℓ ) +∆ℓ |∇v|2 t t t t t t t t nh i h i h i 2 +Bv2 + −2σℓ v +2∇ℓ·∇v dt+σ2θ2ℓ (du )2, t t t t (cid:16) (cid:17) o 10

Unique Continuation for Stochastic Hyperbolic Equations

Similar Unique Continuation for Stochastic Hyperbolic Equations